Why is this object facing the wrong way when exporting to three.js?
Nothing's necessarily wrong, your object just isn't going to be in the same position in ThreeJS as in Blender because the default orientation of the axes is different.

Now, the export system takes care of this. I suspect that your object rotated to make up the difference perfectly, and that the problem is in fact your camera! The same values will cause a different rotation in ThreeJS than in Blender.

You could go to the trouble of manually adjusting to your camera to compensate, but fortunately that isn't necessary. Blender and ThreeJS both have built-in "lookat" functions (here and here, respectively), so you can just export another object to orient them to accordingly (though Blender looks at the entire object while ThreeJS focuses on a single vertex, so make sure the object you use is a small one if you want precision!).